Electromagnetic Fields due to Accelerating Charge

[Yoni V]

Homework Statement

What are the electric and magnetic fields due to a charge that is moving with uniform acceleration?
(Non relativistic)

Homework Equations

Retarded solutions for the vector and scalar potentials.

The Attempt at a Solution

My attempt might be an overkill because I'm using the integral solution
$$
\psi\left(\boldsymbol{r},t\right)=\int_{V}\frac{\left[f\left(\boldsymbol{r}',t\right)\right]_{ret}}{\left|\boldsymbol{r}-\boldsymbol{r}'\right|}d^{3}r'
$$
for the potential as an exercise for correct use of this formula. Perhaps since it's a single particle I can solve for, say, the scalar potential directly from Coulomb's law and just retard the solution. But I'm not sure this is a valid approach because the law essentially assumes electrostatics and immediate information flow without considering the propagation of information through the wave equation (although retarding the solution might just give it the required fix. If anyone can clarify this I'll be grateful). But more importantly, using the above formula I get really mixed up about which of the term is which, and specifically which gets retarded, so I want to be sure I can handle this equation properly for future problems as well.
So I proceed as follows for the scalar potential:

The charge distribution is given by (for simplicity assuming everything is on the x axis):
$$
\rho\left(\boldsymbol{r},t\right)=q\delta\left(y\right)\delta\left(z\right)\delta\left(x-\frac{1}{2}at^{2}\right)
$$
yielding
$$
\phi\left(\boldsymbol{r},t\right)=\int\frac{q\delta\left(y'\right)\delta\left(z'\right)\delta\left(x'-\frac{1}{2}at'^{2}\right)}{\left|\boldsymbol{r}-\boldsymbol{r}'\right|}\delta\left(t'-\left(t-\frac{\left|\boldsymbol{r}-\frac{1}{2}at^{2}\hat{x}\right|}{c}\right)\right)dt'd^{3}r'
\\
=\int\frac{q\delta\left(y'\right)\delta\left(z'\right)\delta\left(x'-\frac{1}{2}a\left(t-\frac{\left|\boldsymbol{r}-\frac{1}{2}at^{2}\hat{x}\right|}{c}\right)^{2}\right)}{\left|\boldsymbol{r}-\boldsymbol{r}'\right|}d^{3}r'
\\
=\frac{q}{\left|\boldsymbol{r}-\frac{1}{2}a\left(t-\frac{\left|\boldsymbol{r}-\frac{1}{2}at^{2}\hat{x}\right|}{c}\right)^{2}\hat{x}\right|}
$$

As can be seen, it really does turn out (at least in this case) just a retardation to the simple solution, but I'm still not sure that I handled the tagging of the variables and retarding them properly.

Is this a correct result? And at which conditions if any may I directly use electrostatics laws such as Coulomb's and retard them in a straightforward fashion? Thanks.

 

[mark726]

Your approach seems correct, although it may be a bit overcomplicated for this particular problem. In general, the electric and magnetic fields due to a charge moving with uniform acceleration can be found using the Lienard-Wiechert potentials, which are the retarded solutions for the vector and scalar potentials. These potentials take into account the propagation of information through the wave equation, and are valid for non-relativistic as well as relativistic speeds.

To answer your question about when you can use electrostatics laws and simply retard them, it depends on the specific problem. In general, if the source of the electric and magnetic fields is moving with a constant velocity, you can use the retarded solutions for the potentials. However, if the source is accelerating, as in this case, you will need to use the Lienard-Wiechert potentials to properly account for the time delay in the propagation of information.

In summary, your approach is correct, but it may be simpler to use the Lienard-Wiechert potentials directly instead of using the integral solution. Additionally, for future problems involving charges moving with uniform acceleration, you should use the Lienard-Wiechert potentials to properly account for the time delay in the propagation of information.